#139423

order	512
nilpotency class	4
isoclinism	isoclinic class of #139360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

353 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,9,6,8,15,3)(4,11,13,16,12,7,14,10)(18,23,26,29)(19,22)(20,28)(24,30)(25,32,27,31);(1,3,5,8,6,9,15,2)(4,10,14,11,12,16,13,7)(17,18,24,32,19,25,30,23,21,26,28,31,22,27,20,29);(1,4,6,12)(2,7,8,11)(3,10,9,16)(5,14,15,13)(17,19,21,22)(18,25,26,27)(20,24,30,28)(23,31,29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=h,d^2=ghi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,f^c=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46619);